Here, I am going to look at a couple of exams that use the Stephens formalism in their consideration of data collected from strained samples. The examples will all be orthorhombic, so that the relevant Stephens parameters are S_{400}, S_{040}, S_{004}, S_{220}, S_{202} and S_{022}.

**\alpha-uranium**

I will start with a study of the microstructural strain energy in \alpha-uranium, by Manley *et al.*, published in Physical Review B. This is orthorhombic with space group *Cmcm*. The study was carried out on polycrystalline samples, collecting neutron powder diffraction patterns from 77 K to 300 K. The authors extract the Stephens parameters from their fits, and later give 3D representations of what they call the microstrain, generated from the Stephens parameters – an example is shown below.

Unfortunately, the actual parameters extracted from the fits are not given in the paper. They also found that they the measured parameters were different in different detector banks, pointing to textural changes in the sample.

They then compared the directionality of the strain broadening with both the Young’s modulus data availables and the hydrostatic compressibility, finding that the pattern is closer to the Young’s modulus state, indicating that the observations more closely mimic a uniaxial state rather than a hydrostatic state.

There then follows an interesting discussion on how to evaluate the strain energy stored in the microstructure. As noted previously, it is not possible to extract values for the stiffness constants directly from the Stephens parameters (note that an orthorhombic lattice has 9 independent stiffness constants, three pure normal stresses, three pure shear stresses, and three coupled terms). They then consider two limits. In the first, the strains are statistically isotropic, such that there are no correlations and the three coupled terms are assumed to be zero. The second limit considers the cases where the stresses are statistically isotropic. In this case, there will be contributions to the strain in one direction, and stress in another through the Poisson effect, and characterized by the Poisson ratio. In reality, the result will lie somewhere between these two limits.

**Pr _{0.5}Ca_{0.5}Mn_{0.97}Ga_{0.03}O_{3}**

This material was being investigated as a colossal magnetoresistive manganite, by Yaicle and co-authors, in a paper published in the Journal of Solid State Chemistry. Some synchrotron X-ray powder diffraction was done, where there was significant broadening of some Bragg peaks, and so the Stephens formalism was used to characterise this. Of the six parameters, four of them were quite large, indicating that the problem could not be simplified to a consideration of only one dimension. However, changes in the strain parameters appeared to correlate with the onset of a charge ordered/orbital ordered state. We can make the same visualisation plots as Manley *et al.* used to see how the response changes.

This makes clear the changes in the strain parameters between 300 K and 10 K, where the two phases have quite different levels of strain – in the Phase I case, the scale is 5 times larger than the other two, indicating the dramatic difference between Phase I and Phase II.